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February 2013 Convergence of latent mixing measures in finite and infinite mixture models
XuanLong Nguyen
Ann. Statist. 41(1): 370-400 (February 2013). DOI: 10.1214/12-AOS1065

Abstract

This paper studies convergence behavior of latent mixing measures that arise in finite and infinite mixture models, using transportation distances (i.e., Wasserstein metrics). The relationship between Wasserstein distances on the space of mixing measures and $f$-divergence functionals such as Hellinger and Kullback–Leibler distances on the space of mixture distributions is investigated in detail using various identifiability conditions. Convergence in Wasserstein metrics for discrete measures implies convergence of individual atoms that provide support for the measures, thereby providing a natural interpretation of convergence of clusters in clustering applications where mixture models are typically employed. Convergence rates of posterior distributions for latent mixing measures are established, for both finite mixtures of multivariate distributions and infinite mixtures based on the Dirichlet process.

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XuanLong Nguyen. "Convergence of latent mixing measures in finite and infinite mixture models." Ann. Statist. 41 (1) 370 - 400, February 2013. https://doi.org/10.1214/12-AOS1065

Information

Published: February 2013
First available in Project Euclid: 26 March 2013

zbMATH: 1347.62117
MathSciNet: MR3059422
Digital Object Identifier: 10.1214/12-AOS1065

Subjects:
Primary: 62F15 , 62G05
Secondary: 62G20

Keywords: $f$-divergence , Bayesian nonparametrics , Dirichlet processes , hierarchical models , mixture distributions , rates of convergence , transportation distances , Wasserstein metric

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 1 • February 2013
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